Chapter 1: Q4P (page 29)

Write the Maclaurin series for $\frac{1}{\sqrt{1 + x}}$ in $\sum$ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example $2$ above

Short Answer

Expert verified

The Maclaurin series is $(\frac{- 1}{2}) = \frac{(- 1)^{n} (2 n - 1)!!}{(2 n)!!}$

Step by step solution

Given Information

Thebinomial series.

Definition of the binomial series.

The Taylor series for the function given by is the binomial series, where is an arbitrary complex number.

Prove the statement.

The binomial series states that $(1 + x)^{p} = \sum_{n = 0}^{\infty} (\begin{array}{l} p \\ n \end{array}) x^{n}$

$(1 + x)^{- 1 / 2} = \sum_{n = 0}^{\infty} (\begin{matrix} - 1 \\ n \end{matrix}) x^{n}$

$= 1 - \frac{1}{2} x + \frac{(\frac{- 1}{2}) (\frac{- 1}{2} - 1)}{2!} x^{2} + \frac{(\frac{- 1}{2}) (\frac{- 1}{2} - 1) (\frac{- 1}{2} - 2)}{3!} x^{3} + \frac{(\frac{- 1}{2}) (\frac{- 1}{2} - 1) (\frac{- 1}{2} - 2) (\frac{- 1}{2} - 3)}{4!} x^{4} + \dots$

$= 1 - \frac{1}{2} x + \frac{(\frac{- 1}{2}) (\frac{- 3}{2})}{2!} x^{2} + \frac{(\frac{- 1}{2}) (\frac{- 3}{2}) (\frac{- 5}{2})}{3!} x^{3} + \frac{(\frac{- 1}{2}) (\frac{- 3}{2}) (\frac{- 5}{2}) (\frac{- 7}{2})}{4!} x^{4} + \dots$

$= 1 - \frac{1}{2} x + \frac{3}{8} x^{2} - \frac{5}{16} x^{3} + \frac{35}{128} x^{4} + \dots$

The formula states that $(\begin{array}{l} p \\ n \end{array}) = \frac{p (p - 1) (p - 2) \dots (p - n + 1)}{n!}$

For $n \geq 2$ , the formula becomes as follows.

$(\begin{array}{l} \frac{- 1}{2} \\ n \end{array}) = \frac{(\frac{- 1}{2}) (\frac{- 1}{2} - 1) (\frac{- 1}{2} - 2) (\frac{- 1}{2} - 3) (\frac{- 1}{2} - (n - 1))}{n!}$

$= \frac{(\frac{- 1}{2}) (\frac{- 3}{2}) (\frac{- 5}{2}) (\frac{- 7}{2}) (\frac{1}{2} - n)}{n!}$

$(\begin{matrix} \frac{- 1}{2} \\ n \end{matrix}) = \frac{(- 1)^{n} 1 \cdot 3 \cdot 5 \cdot 7 \dots (2 n - 1)}{2^{n} n!}$

$(\begin{matrix} \frac{- 1}{2} \\ n \end{matrix}) = \frac{(- 1)^{n} (2 n - 1)!!}{(2 n)!!}$

The statement has been proven.

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Write the Maclaurin series for $\frac{1}{\sqrt{1 + x}}$ in $\sum$ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example $2$ above

Short Answer

Step by step solution

Given Information

Definition of the binomial series.

Prove the statement.

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Write the Maclaurin series for 11+xin∑ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example 2 above

Short Answer

Step by step solution

Given Information

Definition of the binomial series.

Prove the statement.

One App. One Place for Learning.

Most popular questions from this chapter

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Write the Maclaurin series for $\frac{1}{\sqrt{1 + x}}$ in $\sum$ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example $2$ above